#192958

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(17,18)(19,28)(20,25)(21,26)(22,27)(23,30)(24,29)(31,32);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,23)(20,29)(21,30)(22,31)(24,27)(25,28)(26,32);(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,32)(26,28);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(17,20,22,21)(18,25,27,26)(19,29,31,30)(23,28,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100154);